On this day six years ago, high-tech got a godsend: Google went public.It started at $85 a share and doubled in 67 days. The stock doubled again in 12 months, and it had doubled a third time by two years later.

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That’s up eight-fold in three years and two months, if you do the math. And then, on November 6, 2007, Google hit its all-time closing-price high of $741.79.

But lately Google trades below the $500 mark. The Dow, soaring off the lows of March 2009, still is down 26 percent from its high in October ’07; Google stubbornly remains 35 percent below the high it reached the same month.

Do investors have it wrong—or is Google itself doing something wrong? My answer would be . . . yes.

By sheer numbers a higher price should apply to the world’s dominant search engine on a global network that, in a few years, could link a trillion devices. In three years since hitting that high, Google has doubled revenue, almost doubled already-prodigious profits, more than doubled its cash on hand (to $30 billion!) and doubled its total assets.

It’s not enough. That, at least, is what investors are saying. Today Google trades at 21 times trailing earnings, about where the broader market lies. But isn’t this company worth more, pound for pound, than the typical S&P old-guarder?

A few years ago a scary-smart tech consultant gave me a good rap on why Google, in five or ten years, might grow to $1,000 a share.

For one of the first times in the history of advertising, this guy observed, Google brought together the ultimate dream for marketers: It could deliver an ad that customers themselves sought out, for precisely the product they were shopping for at that moment, in a venue that let them buy it instantly, right then and there.

Remember that old saying from the 1920s retailer, John Wanamaker? He knew half his ad budget was wasted, now if only he knew which half. Google promised to change all that.

This approach served the company remarkably well. As 2007 ended, Google had full-year revenue of almost $10 billion; net income of $4.2 billion; $14.3 billion in cash, and total assets north of $25 billion. This, for a company less than a decade old.

Now, just three years later, Google is likely to have revenue of $26 billion; net income exceeding $7 billion; $30 billion in cash, and total assets of almost $50 billion.

The biggest reason for Google stock’s ennuiis the same, old same-old for any maturing tech superstar: Growth is slowing. The Law of Large Numbers. Adding resistance to Google’s rise are the global economy’s near-death experience, advertising’s role as the first thing to get cut in a downturn, and investor fickleness.

But the company itself also may be to blame. Google is driven by brilliance and hubris and attention-deficit hyperactivity disorder. Instead of focusing ever harder on its core—providing smarter search results and better-tailored ads for what you’re looking for, thereby boosting the only real revenue stream the company has—Google has wanderlust.

Giving Napoleon the finger, Google fights wars on myriad fronts all at once. Its Android software in smartphones takes aim at Apple’s iPhone. It offers free software apps and Chrome OS, undercutting Microsoft’s business for no other real purpose than just to mess with them.

Google also is tackling Apple iTunes in online music, trying to bear-hug (or maul) the broadcast networks and cable channels in Net-tv, and meddling in books on-shelf and online. Ever omnivorous, it’s itching to get into Facebook’s face in social networking.

The conceit of all this activity is that it will spur more searches and more ad-clicks, with Google collecting more revenue. (My suspicion: The real motivator is that these guys would get bored fast if search were their only sandbox; that, or they already did.)

Sun Microsystems made similar claims that its free Java programming language would drive sales of Sun servers, in the years before it faltered and ultimately faded into Oracle.

Google’s case is far more convincing, especially as it girds for the next online ad boom: extremely local real-time ads to your GPS smartphone as you Net-surf and text with friends.

Long-term the opportunity is enormousfor this already-giant company. Online ads are still only 10 percent or so of all ad spending, online retail sales are at a similar share of all retail—couldn’t both businesses double to 20 percent of the pie or more by, say, late this decade?

But in the near term, investors need to see the potential for higher growth now, and that leads them back to Google’s raison d’etre: search ads and how much marketers are willing to spend on them.

The Google guys don’t seem to rhapsodize much these days about the frontiers of perfect search, not when they have platforms to fight for and wars to wage with Apple and Microsoft.

I’d love to see Google create an entirely new breakthrough in display ads online; or make video a mainstay of search ads; or forge a new people-search “vertical” for missing or estranged loved ones; or even devise a system letting millions of people place instant personal ads aimed at only a few dozen like-minded neighbors. You know, search-ad stuff, Google’s real business?

And wouldn’t it be great if searches got more surgical, more anticipative and adaptive, more intuitive and useful? The other day I googled up the latest on a hotel hotspot in New York and was offered reservations for a similarly named outpost. In Switzerland. Now, that’s just . . . silly.